{"id":803,"date":"2016-08-25T11:11:14","date_gmt":"2016-08-25T16:11:14","guid":{"rendered":"https:\/\/trigonography.com\/?p=803"},"modified":"2022-06-25T04:29:19","modified_gmt":"2022-06-25T09:29:19","slug":"chaikovskys-involute-pinwheel","status":"publish","type":"post","link":"https:\/\/trigonography.com\/?p=803","title":{"rendered":"Chaikovsky’s Involute Pinwheel: Power Series for Sine and Cosine"},"content":{"rendered":"\n

Attributed to mathematician and teacher Y. S. Chaikovsky, circa 1935.
(Gurin, Leo S.\u2003”A Problem.”\u2003The American Mathematical Monthly<\/cite> 103, no. 8 (1996): 683-86. JSTOR<\/a>)<\/p>\n\n\n

\n
\"trigonograph-sincosinvolutes\"<\/a><\/figure>\n<\/div>\n\n\n

\\(\\stackrel{\\frown}{PP}_1\\) is an arc of the unit circle; thereafter, \\(\\stackrel{\\frown}{PP}_i\\) is an involute<\/a> of \\(\\stackrel{\\frown}{PP}_{i-1}\\).<\/p>\n\n\n\n

$$
\\begin{align}
\\cos\\theta &\\;=\\; |\\overline{P_0P_1}| \\;-\\; |\\overline{P_2 P_3}| \\;+\\; |\\overline{P_4P_5}| \\;-\\; \\cdots \\\\[4pt]
&\\;=\\; |\\overline{P_0P}| \\;-\\; |\\stackrel{\\frown}{P_2 P}| \\;+\\; |\\stackrel{\\frown}{P_4P}| \\;-\\; \\cdots \\\\[4pt]
&\\;=\\; \\sum_{k=0}^{\\infty} \\frac{(-1)^{k}}{(2k)!}\\;\\theta^{2k} \\\\[12pt]
\\sin\\theta &\\;=\\; |\\overline{P_1P_2}| \\;-\\; |\\overline{P_3 P_4}| \\;+\\; |\\overline{P_5P_6}| \\;-\\; \\cdots \\\\[4pt]
&\\;=\\; |\\overline{P_1P}| \\;-\\; |\\stackrel{\\frown}{P_3 P}| \\;+\\; |\\stackrel{\\frown}{P_5P}| \\;-\\; \\cdots \\\\[4pt]
&\\;=\\; \\sum_{k=0}^{\\infty} \\frac{(-1)^k}{(2k+1)!}\\;\\theta^{2k+1}
\\end{align}$$<\/div>\n\n\n\n

While the trigonograph makes the convergence<\/em> of the series clear, the fact that
the involute lengths are appropriately-scaled powers of \\(\\theta\\) is decidedly not.
Gurin’s article explains Chaikovsky’s clever use of of combinatorics
(and a fundamental limit from Calculus) to prove the connection, but
one may also treat this as an exercise in parametric arc length and recursion.<\/p>\n\n\n\n

Get the t-shirt … or the other one!<\/b>
\n<\/a>\n  <\/a><\/div>\n","protected":false},"excerpt":{"rendered":"

Attributed to mathematician and teacher Y. S. Chaikovsky, circa 1935.(Gurin, Leo S.\u2003”A Problem.”\u2003The American Mathematical Monthly 103, no. 8 (1996): 683-86. JSTOR) \\(\\stackrel{\\frown}{PP}_1\\) is an arc of the unit circle; thereafter, \\(\\stackrel{\\frown}{PP}_i\\) is an involute of \\(\\stackrel{\\frown}{PP}_{i-1}\\). $$\\begin{align}\\cos\\theta &\\;=\\; |\\overline{P_0P_1}| \\;-\\; |\\overline{P_2 P_3}| \\;+\\; |\\overline{P_4P_5}| \\;-\\; \\cdots \\\\[4pt]&\\;=\\; |\\overline{P_0P}| \\;-\\; |\\stackrel{\\frown}{P_2 P}| \\;+\\; |\\stackrel{\\frown}{P_4P}| \\;-\\; […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-803","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"_links":{"self":[{"href":"https:\/\/trigonography.com\/index.php?rest_route=\/wp\/v2\/posts\/803","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/trigonography.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/trigonography.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/trigonography.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/trigonography.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=803"}],"version-history":[{"count":9,"href":"https:\/\/trigonography.com\/index.php?rest_route=\/wp\/v2\/posts\/803\/revisions"}],"predecessor-version":[{"id":1249,"href":"https:\/\/trigonography.com\/index.php?rest_route=\/wp\/v2\/posts\/803\/revisions\/1249"}],"wp:attachment":[{"href":"https:\/\/trigonography.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=803"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/trigonography.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=803"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/trigonography.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=803"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}